Cho $\triangle{ABC}$ cân tại $A$. Vẽ $AH \perp BC \; (H\in BC)$. Gọi $M, N$ lần lượt là trung điểm của $AC, AB$.
Vẽ $MK\perp BC \; (K \in BC)$, $NH$ cắt $BM$ tại $O$, trung trực của $MH$ cắt $MK$ tại $I$.
Chứng minh : $\widehat{AOI} = 90^\circ$.
*** Cannot compile formula: \definecolor{qqwuqq}{rgb}{0.,0.39215686274509803,0.}\definecolor{zzttqq}{rgb}{0.6,0.2,0.}\definecolor{qqqqff}{rgb}{0.,0.,1.}\begin{tikzpicture}[line cap=round,line join=round,>=triangle 45,x=1.0cm,y=1.0cm]\clip(0.,0.) rectangle (8.,9.);\fill[color=zzttqq,fill=zzttqq,fill opacity=0.1] (4.,8.) -- (1.,1.) -- (7.,1.) -- cycle;\draw[color=qqwuqq,fill=qqwuqq,fill opacity=0.1] (5.5,1.3902090920140011) -- (5.109790907985999,1.3902090920140011) -- (5.109790907985999,1.) -- (5.5,1.) -- cycle; \draw[color=qqwuqq,fill=qqwuqq,fill opacity=0.1] (4.,1.3902090920140011) -- (3.6097909079859987,1.3902090920140011) -- (3.6097909079859987,1.) -- (4.,1.) -- cycle; \draw [color=zzttqq] (1.,1.)-- (7.,1.);\draw (2.5,4.5)-- (4.,1.);\draw (1.,1.)-- (5.5,4.5);\draw (5.5,4.5)-- (4.,1.);\draw (3.25,2.75)-- (5.5,2.4285714285714284);\draw (3.25,2.75)-- (4.,8.);\draw (4.,8.)-- (5.5,2.4285714285714284);\draw (5.5,2.4285714285714284)-- (4.,1.);\draw (4.83039723217654,1.6765450017452488) -- (4.716223648020506,1.796427265109081);\draw (4.783776351979493,1.6321441634623457) -- (4.669602767823459,1.7520264268261776);\draw (3.25,2.75)-- (4.75,5.214285714285714);\draw (5.5,4.5)-- (5.5,2.4285714285714284);\draw (5.582775848513124,3.4964763220408184) -- (5.417224151486876,3.4964763220408184);\draw (5.582775848513124,3.4320951065306113) -- (5.417224151486876,3.4320951065306113);\draw (5.5,2.4285714285714284)-- (5.5,1.);\draw [color=zzttqq] (4.,8.)-- (2.5,4.5);\draw [color=zzttqq] (3.326083009766366,6.217392995814415) -- (3.1739169902336353,6.282607004185587);\draw [color=zzttqq] (4.,8.)-- (5.5,4.5);\draw [color=zzttqq] (4.826083009766366,6.282607004185587) -- (4.673916990233633,6.217392995814415);\draw [color=zzttqq] (5.5,4.5)-- (7.,1.);\draw [color=zzttqq] (6.326083009766367,2.7826070041855844) -- (6.173916990233634,2.7173929958144156);\draw [color=zzttqq] (2.5,4.5)-- (1.,1.);\draw [color=zzttqq] (1.826083009766365,2.7173929958144143) -- (1.6739169902336344,2.7826070041855853);\draw (4.,8.)-- (4.,1.);\begin{scriptsize}\draw [fill=qqqqff] (4.,8.) circle (1.5pt);\draw[color=qqqqff] (4.134468576086929,8.24987566530852) node {\textsf{A}};\draw [fill=qqqqff] (1.,1.) circle (1.5pt);\draw[color=qqqqff] (0.8418292685648975,1.2967043902061428) node {\textsf{B}};\draw [fill=qqqqff] (7.,1.) circle (1.5pt);\draw[color=qqqqff] (7.132793755562299,1.25991512420031) node {\textsf{C}};\draw [fill=qqqqff] (4.,1.) circle (1.5pt);\draw[color=qqqqff] (3.7481812830256858,1.2231258581944773) node {\textsf{H}};\draw [fill=qqqqff] (5.5,4.5) circle (1.5pt);\draw[color=qqqqff] (5.624433849323156,4.754895394754415) node {\textsf{M}};\draw [fill=qqqqff] (2.5,4.5) circle (1.5pt);\draw[color=qqqqff] (2.423767706815706,4.846868559768997) node {\textsf{N}};\draw [fill=qqqqff] (5.5,1.) circle (1.5pt);\draw[color=qqqqff] (5.624433849323156,1.25991512420031) node {\textsf{K}};\draw [fill=qqqqff] (3.25,2.75) circle (1.5pt);\draw[color=qqqqff] (3.1595530269323615,2.5659340674073703) node {\textsf{O}};\draw [fill=qqqqff] (5.5,2.4285714285714284) circle (1.5pt);\draw[color=qqqqff] (5.624433849323156,2.694696498427785) node {\textsf{I}};\draw [fill=qqqqff] (4.75,5.2143) circle (1.5pt);\draw[color=qqqqff] (4.87,5.47) node {\textsf{E}};\end{scriptsize}\end{tikzpicture} *** Error message: Error: Nothing to show, formula is empty